Lie algebra actions on module categories for truncated shifted Yangians

TL;DR

This paper develops a categorical framework for Coulomb branch algebras, generalizing induction and restriction functors, and establishes a categorification of the geometric Satake correspondence via flavoured KLRW algebras.

Contribution

It introduces a new class of flavoured KLRW algebras and constructs a categorical action of Kac-Moody algebras on Coulomb branch modules, extending prior work to new algebraic structures.

Findings

Defined parabolic induction and restriction functors for Coulomb branch algebras.

Established an equivalence between Gelfand-Tsetlin modules and flavoured KLRW modules.

Provided a categorification of the geometric Satake correspondence for truncated shifted Yangians.

Abstract

We develop a theory of parabolic induction and restriction functors relating modules over Coulomb branch algebras, in the sense of Braverman-Finkelberg-Nakajima. Our functors generalize Bezrukavnikov-Etingof's induction and restriction functors for Cherednik algebras, but their definition uses different tools. After this general definition, we focus on quiver gauge theories attached to a quiver $\Gamma$. The induction and restriction functors allow us to define a categorical action of the corresponding symmetric Kac-Moody algebra $\mathfrak{g}_{\Gamma}$ on category $ \mathcal O $ for these Coulomb branch algebras. When $ \Gamma $ is of Dynkin type, the Coulomb branch algebras are truncated shifted Yangians and quantize generalized affine Grassmannian slices. Thus, we regard our action as a categorification of the geometric Satake correspondence. To establish this categorical action, we define a new class of "flavoured" KLRW algebras, which are similar to the diagrammatic algebras originally constructed by the second author for the purpose of tensor product categorification. We prove an equivalence between the category of Gelfand-Tsetlin modules over a Coulomb branch algebra and the modules over a flavoured KLRW algebra. This equivalence relates the categorical action by induction and restriction functors to the usual categorical action on modules over a KLRW algebra.